Solve system of 5 trigonometric equationsWhat is the best way to solve the system of following trigonometric equations with respect to w l , w r , b l , b r ? F x = w l 2 s i n ( b l ) + w r 2 s i n ( b r ) F z = w l 2 c o s ( b l ) + w r 2 c o s ( b r ) M x = w r 2 c o s ( b r ) − w l 2 c o s ( b l ) M y = w l 2 s i n ( b l ) + w r 2 s i n ( b r ) M z = w r 2 s i n ( b r ) − w l 2 s i n ( b l )I need to find relation between w l , w r , b l , b r and symbolic coefficients F x , F z , M x , M y , M z .

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Solve system of 5 trigonometric equationsWhat is the best way to solve the system of following trigonometric equations with respect to       w    l    ,      w    r    ,      b    l    ,      b    r  ?      F    x    =      w    l    2    s  i  n  (      b    l    )  +      w    r    2    s  i  n  (      b    r    )      F    z    =      w    l    2    c  o  s  (      b    l    )  +      w    r    2    c  o  s  (      b    r    )      M    x    =      w    r    2    c  o  s  (      b    r    )  −      w    l    2    c  o  s  (      b    l    )      M    y    =      w    l    2    s  i  n  (      b    l    )  +      w    r    2    s  i  n  (      b    r    )      M    z    =      w    r    2    s  i  n  (      b    r    )  −      w    l    2    s  i  n  (      b    l    )I need to find relation between       w    l    ,      w    r    ,      b    l    ,      b    r   and symbolic coefficients       F    x    ,      F    z    ,      M    x    ,      M    y    ,      M    z  .

Eynardfb0 · Accepted Answer

Note that      F    z    +      M    x    =  2      w    r    2    cos  ⁡      b    r        F    x    +      M    z    =  2      w    r    2    sin  ⁡      b    r  thus for       b    r   and       w    r   we have (dividing one equation by another, using       sin    2    ⁡  θ  +      cos    2    ⁡  θ  =  1:  tan  ⁡      b    r    =                    F        x            +              M        z                            F        z            +              M        x              4      w    r    4    =  (      F    x    +      M    z        )    2    +  (      F    z    +      M    x        )    2  Note also that      F    z    −      M    x    =  2      w    l    2    cos  ⁡      b    l        F    x    −      M    z    =  2      w    1    2    sin  ⁡      b    l  thus for       b    l   and       w    l   we have  tan  ⁡      b    l    =                    F        x            −              M        z                            F        z            −              M        x              4      w    l    4    =  (      F    x    −      M    z        )    2    +  (      F    z    −      M    x        )    2  (As mentioned in the comments by OP,       M    y   turns out to be unused indeed.)

Kale Sampson · Accepted Answer

If you ignore the fourth equation, since       F    x   and       M    y   are identical, then number the remaining equations 1-4:Take successively the sum of equations 1 and 4,and the difference of 1 and 4.Call these equations A and B. Do the same operations on equations 2 and 3, call these C and D.Now dividing B and D gives  tan  ⁡      β    l    =                    F        x            −              M        z                            F        z            −              M        x            Squaring and adding the same gives  4      ω    l    4    =  (      F    x    −      M    z        )    2    +  (      F    z    −      M    x        )    2  Now do the same set of operations on equations A and C and you&#039;re done.

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